I was thinking warm water and dishwashing liquid (Dawn?) and maybe a scrub brush to help with the lead. Agree on not doing it in the dishwasher. I probably wouldn't do it in the kitchen sink either, but a bucket, utility pan, or utility sink instead.
For the most part, your CLEAN Factory, or reloaded ammo will not dirty the ammo boxes.
They get real nasty when you put your fired cases back in them..Take yourself a Wally World bag or coffee can for your fired brass and put your nice boxes back into your range bag for the next round of CLEAN ammo...Bill.